For any non-zero natural integer $n$, we set
$$\omega(n) = \operatorname{Card}\{p \text{ prime} : p \mid n\} = \sum_{\substack{p \mid n \\ p \text{ prime}}} 1.$$
Show that
$$\sum_{n \leqslant x} \omega(n)^{2} = \sum_{\substack{p_{1} \leqslant x \\ p_{1} \text{ prime}}} \sum_{\substack{p_{2} \leqslant x \\ p_{2} \text{ prime}}} \operatorname{Card}\left\{n \in \mathbb{N}^{*} : n \leqslant x, p_{1} \mid n \text{ and } p_{2} \mid n\right\}$$